Error analysis of indirect force determination and a regularisation method to reduce force determination error

Accurate estimation of dynamic forces acting on a structure is very difficult, since, at certain frequency regions, the identified forces may contain an unacceptable amount of error. In this study, by investigating the characteristics of the inverse transfer function and frequency response function (FRF) error, the indirect force determination error is analysed statistically, and the special frequency regions where force determination error is very large are searched for each case of the number of response measurements and applied forces. It is shown that the large force determination error near the first kind of inverse pole frequencies is mainly due to the rank deficiency of the FRF submatrix, while the large error in the vicinity of resonance frequencies comes from the larger FRF error. The idea was tested both analytically and experimentally. Also a regularisation process to reduce the error especially near the first kind of inverse pole frequencies is proposed. Since the degree of singularity of the FRF submatrix near the first kind of inverse pole frequencies is mainly related to system damping, a regularisation procedure is suggested by adding additional damping. An optimal regularisation constant is also derived. It is shown that, the greater the order of magnitude of the FRF error is than that of modal damping ratio, the larger the force determination error reduction is obtained with the suggested regularisation procedure. The proposed regularisation method was tested experimentally and its effects examined.